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Foundations of Tensor Analysis for Students of Physics and Engineering with an Introduction to the Theory of Relativity

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Foundations of Tensor Analysis for Students of Physics and Engineering with an Introduction to the Theory of Relativity NASA/TP—2005-213115 Foundations of Tensor Analysis for Students of Physics and Engineering With an Introduction to the Theory of Relativity Joseph C. Kolecki Glenn Research Center, Cleveland, Ohio April 2005 The NASA STI Program Office . in Profile Since its founding, NASA has been dedicated to • CONFERENCE PUBLICATION. Collected the advancement of aeronautics and space papers from scientific and technical science. The NASA Scientific and Technical conferences, symposia, seminars, or other Information (STI) Program Office plays a key part meetings sponsored or cosponsored by in helping NASA maintain this important role. NASA. The NASA STI Program Office is operated by • SPECIAL PUBLICATION. Scientific, Langley Research Center, the Lead Center for technical, or historical information from NASA’s scientific and technical information. 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Kolecki Glenn Research Center, Cleveland, Ohio National Aeronautics and Space Administration Glenn Research Center April 2005 Acknowledgments To Dr. Ken DeWitt of Toledo University, I extend a special thanks for being a guiding light to me in much of my advanced mathematics, especially in tensor analysis. Years ago, he made the statement that in working with tensors, one must learn to find—and feel—the rhythm inherent in the indices. He certainly felt that rhythm, and his ability to do so made a major difference in his approach to teaching the material and enabling his students to comprehend it. He read this work and made many valuable suggestions and alterations that greatly strengthened it. I wish to also recognize Dr. Harold Kautz’s contribution to the section Magnetic Permeability and Material Stress, which was derived from a conversation with him. Dr. Kautz has been my colleague and part-time mentor since 1973. Available from NASA Center for Aerospace Information National Technical Information Service 7121 Standard Drive 5285 Port Royal Road Hanover, MD 21076 Springfield, VA 22100 Available electronically at http://gltrs.grc.nasa.gov Contents Summary ............................................................................................................................................ 1 Introduction ........................................................................................................................................ 1 Alegbra ............................................................................................................................................... 1 Statement of Core Ideas ............................................................................................................... 1 Number Systems .......................................................................................................................... 2 Numbers, Denominate Numbers, and Vectors............................................................................. 3 Formal Presentation of Vectors.................................................................................................... 3 Vector Arithmetic ........................................................................................................................ 5 Dyads and Other Higher Order Products ..................................................................................... 8 Dyad Arithmetic........................................................................................................................... 10 Components, Rank, and Dimensionality...................................................................................... 13 Dyads as Matrices ........................................................................................................................ 14 Fields............................................................................................................................................ 15 Magnetic Permeability and Material Stress ................................................................................. 16 Location and Measurement: Coordinate Systems........................................................................ 18 Multiple Coordinate Systems: Coordinate Transformations........................................................ 19 Coordinate Independence............................................................................................................. 20 Coordinate Independence: Another Point of View ...................................................................... 21 Coordinate Independence of Physical Quantities: Some Examples............................................. 23 Metric or Fundamental Tensor..................................................................................................... 24 Coordinate Systems, Base Vectors, Covariance, and Contravariance ......................................... 27 Kronecker’s Delta and the Identity Matrix .................................................................................. 29 Dyad Components: Covariant, Contravariant, and Mixed........................................................... 30 Relationship Between Covariant and Contravariant Components of a Vector............................ 30 st s Relation Between gij, g , and δw ................................................................................................. 32 Inner Product as an Operation Involving Mixed Indices ............................................................. 32 General Mixed Component: Raising and Lowering Indices........................................................ 34 Tensors: Formal Definitions ........................................................................................................ 35 Is the Position Vector a Tensor? .................................................................................................. 38 The Equivalence of Coordinate Independence With the Formal Definition for a Rank 1 Tensor (Vector)......................................................................................................... 39 Coordinate Transformation of the Fundamental Tensor and Kronecker’s Delta......................... 40 Two Examples From Solid Analytical Geometry........................................................................ 40 Calculus .............................................................................................................................................. 42 Statement of Core Idea................................................................................................................. 42 First Steps Toward a Tensor Calculus: An Example From Classical Mechanics........................ 42 Base Vector Differentials: Toward a General Formulation ......................................................... 48 Another Example From Polar Coordinates.................................................................................. 50 Base Vector Differentials in the General Case ............................................................................ 51 Tensor Differentiation: Absolute and Covariant Derivatives ...................................................... 55 k Tensor Character of Γwt .............................................................................................................. 56 Differentials of Higher Rank Tensors.......................................................................................... 58 Product
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